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  • A large steel wheel is to be fitted on to a shaft of the same material. At 27 degree C, the outer diameter of the shaft is 8.70 cm and the diameter of the central hole in the wheel is 8.69 cm.

Q. 11.7     A large steel wheel is to be fitted on to a shaft of the same material. At 27^{\circ}C, the outer diameter of the shaft is 8.70\; cm and the diameter of the central hole in the wheel is 8.69\; cm. The shaft is cooled using ‘dry ice’. At what temperature of the shaft does the wheel slip on the shaft? Assume coefficient of linear expansion of the steel to be constant over the required temperature range :  \alpha _{steel}=1.20\times 10^{-5}K^{-1}.

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Diameter of the steel shaft at 27 oC (T1) d1 = 8.70 cm

The diameter of the central hole in the wheel d2 = 8.69 cm

Coefficient of linear expansion of the steel \alpha _{steel}=1.20\times 10^{-5}K^{-1}.

The wheel will slip on the shaft when the diameter of the steel shaft becomes equal to the diameter of the central hole in the wheel.

Let this happen at temperature T

\\d_{2}=d_{1}(1+\alpha (T-T_{1})\)\\ 8.69=8.7(1+1.2\times 10^{5}(T-27))\\ T=-68.79\ ^{o}C

 

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